An alpha male is, to put it in the most simplest way, powerful. That's it--the more powerful a male, the more of an alpha male he is. This power roots from his superior intelligence, as well as his superior physical stature.
People often associate the term "alpha male" with this sort of openly-overconfident male who is an outgoing extrovert. This isn't true. The allpha male would not have to rely on such socialising to attract people; the male who attracts people despite staying out of the spotlight and remaining silent is the real alpha male.
You're walking down the street and cross paths with a fairly muscular guy in dark clothes, looking down, right into your eyes so intimidatingly. He looks so quiet and reserved, yet so confident and definite. That guy was an alpha male and you probably now wanna get ravished by him in bed.